A Doubling Dimension Threshold Θ(log log n )

Laboratoire de l’Informatique du Parall´elisme

Ecole Normale Sup´erieure de Lyon´

Unit´e Mixte de Recherche CNRS-INRIA-ENS LYON-UCBL n^o5668

A Doubling Dimension Threshold Θ(log log n )

for Augmented Graph Navigability

Pierre Fraigniaud Emmanuelle Lebhar Zvi Lotker

Avril 2006

Research Report N^o2006-16

Ecole Normale Sup´erieure de Lyon ´

46 All´ee d’Italie, 69364 Lyon Cedex 07, France T´el´ephone : +33(0)4.72.72.80.37 T´el´ecopieur : +33(0)4.72.72.80.80 Adresse ´electronique :lip@ens-lyon.fr

A Doubling Dimension Threshold Θ(log log n ) for Augmented Graph Navigability

Pierre Fraigniaud Emmanuelle Lebhar

Zvi Lotker Avril 2006

Abstract

In his seminal work, Kleinberg showed how to augment meshes using random edges, so that they become navigable; that is, greedy routing computes paths of polylogarithmic expected length between any pairs of nodes. This yields the crucial question of determining wether such an augmentation is possible for all graphs. In this paper, we answer negatively to this question by exhibiting a threshold on the doubling dimension, above which an infinite family of graphs cannot be augmented to become navigable whatever the distribution of random edges is. Precisely, it was known that graphs of doubling dimension at most O(log logn) are navigable. We show that for doubling dimension log logn, an infinite family of graphs cannot be augmented to become navigable. Finally, we complete our result by studying square meshes, that we prove to always be augmentable to become navigable.

Keywords: doubling dimension, small world, greedy routing.

R´esum´e

Kleinberg a montr´e comment augmenter les grilles par des liens al´eatoires de fa¸con `a ce qu’elles deviennent navigables; c’est-`a-dire que le routage glouton calcule des chemins de longueur polylogarithmique en esp´erance entre toute paire de noeuds. Cela conduit `a la question cruciale de d´eterminer si une telle augmentation est possible pour tout graphe. Dans cet article, nous r´epondons n´egativement `a cette question en exhibant un seuil sur la dimension doublante, au-dessus duquel une famille infinie de graphes ne peut pas ˆetre augment´ee pour devenir navigable, quelle que soit la distribution de liens. Pr´ecis´ement, il

´etait connu que les graphes de dimension doublante au plus O(log logn) sont navigable. Nous montrons que pour une dimension doublantelog logn, une famille infinie de graphes ne peut ˆetre augment´ee pour devenir navigable. Enfin, nous compl´etons notre r´esultat en ´etudiant les grilles que nous d´emontrons pouvoir toujours ˆetre augment´ees pour devenir navigables.

Mots-cl´es: dimension doublante, petit monde, routage glouton.

1 Introduction

Thedoubling dimension [4, 13, 15] appeared recently as a key parameter for measuring the ability of networks to support efficient algorithms [7, 14] or to realize specific tasks efficiently [1, 2, 6, 24]. Roughly speaking, the doubling dimension of a graphGis the smallestdsuch that, for any integerr≥1, and for any nodeu∈V(G), the ballB(u,2r) centered atuand of radius 2rcan be covered by at most 2^d balls B(ui, r) centered at nodesui∈V(G). (This definition can be extended to any metric, and, for instance, Z^d with the `1 norm is of doubling dimension d). In particular, the doubling dimension has an impact on the analysis of the small world phenomenon [22], precisely on the expected performances of greedy routing inaugmented graphs [17].

An augmented graph is a pair (G, ϕ) whereGis ann-node graph, andϕis a collection of probability distributions{ϕu, u∈V(G)}. Every nodeu∈V(G) is given an extra link pointing to some nodev, called thelong range contact of u. The link from a node to its long range contact is called along range link.

The original links of the graph are called local links. The long range contact of uis chosen at random according to ϕu as follows : Pr{u → v} = ϕu(v). Greedy routing in (G, ϕ) is the oblivious routing protocol where the routing decision taken at the current node ufor a message of destinationtconsists in (1) selecting a neighbor v of u that is the closest to t according to the distance in G (this choice is performed among all neighbors of uin G and the long range contact ofu), and (2) forwarding the message tov. This process assumes that every node has a knowledge of the distances inG, or at least a good approximation of them. On the other hand, every node is unaware of the long range links added to G, but its own long range link. Hence the nodes have no notion of the distances in the augmented graph.

Note that the knowledge of the distances in the underlying graphGis a reasonable assumption when, for instance,Gis a network in which distances can be computed from the coordinates of the nodes (e.g., in meshes, as in [17]).

An infinite family of graphsG ={G⁽ⁱ⁾, i∈I}is navigable if there exists a family Φ ={ϕ⁽ⁱ⁾, i∈I} of collections of probability distributions, and a functionf(n)∈O(polylog(n)) such that, for anyi∈I, greedy routing in (G⁽ⁱ⁾, ϕ⁽ⁱ⁾) performs in at most f(n⁽ⁱ⁾) expected number of steps where n⁽ⁱ⁾ is the order of the graphG⁽ⁱ⁾. More precisely, for any pair of nodes (s, t) ofG⁽ⁱ⁾, the expected number of steps E(ϕ⁽ⁱ⁾, s, t) for traveling froms to t using greedy routing in (G⁽ⁱ⁾, ϕ⁽ⁱ⁾) is at most f(n⁽ⁱ⁾). Thegreedy diameter of (G⁽ⁱ⁾, ϕ⁽ⁱ⁾) is defined as max_s,t∈G⁽ⁱ⁾E(ϕ⁽ⁱ⁾, s, t).

In his seminal paper, Kleinberg [17] proved that, for any fixed integer d ≥ 1, the family of d- dimensional meshes is navigable. Duchon et al [8] generalized this result by proving that any infinite family of graphs with bounded growth is navigable. Fraigniaud [11] proved that any infinite family of graphs with bounded treewidth is navigable. Finally, Slivkins [24] recently related navigability to doubling dimension by proving that any infinite family of graphs with doubling dimension at mostO(log logn) is navigable. All these results naturally lead to the question of whether all graphs are navigable.

Letδ:N7→N, letG^n,δ(n)be the class ofn-node graphs with doubling dimension at most δ(n), and letG^δ =∪ⁿ≥1G^n,δ(n). By rephrasing Slivkins result [24], we get thatG^δ is navigable for any function δ bounded from above byclog lognfor some constantc >0. This however lets open the case of graphs of larger doubling dimensions, namely the cases of all familiesG^δ where δis satisfyingδ(n)log logn.

1.1 Our results

We prove a threshold ofδ(n) = Θ(log logn) for the navigability ofG^δ : below a certain function δ, G^δ is navigable, while above it G^δ is not navigable. More precisely, we prove that, for any function δ satisfying limn→∞(log logn)/δ(n) = 0, G^δ is not navigable. Hence, the result in [24] is essentially the best that can be achieved by considering only the doubling dimension of graphs.

Our negative result requires to prove that for an infinite family of graphs inG^δ,any distribution of the long range links leaves the expected number of steps of greedy routing above any polylogarithmic for some pairs of source and target. For this purpose, we exhibit graphs presenting a very high number of possible “directions” for a long range link to go. By a counting argument, we show that there exist pairs of source and target at distance greater than any polylogarithm, between which greedy routing does not use any long range link, whatever their distribution is. In other words, we exhibit an infinite family of graphs with non polylogarithmic greedy diameter for any augmentation. This negative results answers a question asked in [11, 18].

1

We also prove a somehow counter intuitive result by showing that a supergraph of a navigable graph is not necessarily navigable. In particular, we show that all square meshes are navigable, for all dimensions.

Specifically, we prove that although the family of non navigable graphs that we use to disprove the navigability of all graphs contains the standard square meshes of dimensionδ as subgraphs, this latter family of graphs is navigable.

1.2 Related works

Kleinberg showed that greedy routing performs in O(log²n) steps between any pair of nodes on d-dimensional meshes augmented by thed-harmonic distribution. I.e. the greedy diameter of these aug- mented meshes is O(log²n). Since then, several results have been developed to tighten the analysis of greedy routing on randomly augmented networks. Precisely, Barri`ere et al. [5] showed that the greedy diameter of thed-dimensional meshes augmented by thed-harmonic distribution is Θ(log²n). In the spe- cial case of rings, Aspnes et al. [3] proved a lower bound on the greedy diameter of Ω(log²n/log logn) for any augmentation. For paths, Flammini et al. [9] recently showed a lower bound on the greedy diameter of Ω(log²n) in the case of symmetric and distance monotonic augmentations. Martel and Nguyen [21]

showed however that the (standard) diameter of all these networks augmented by the harmonic distri- bution is Θ(logn). In another perspective, several authors developed decentralized algorithms for the d-dimensional mesh augmented by thed-harmonic distribution. Lebhar and Schabanel [19] presented a decentralized algorithm which performs inO(logn(log logn)²) expected number of steps in this graph.

The algorithm Neighbor-Of-Neighbor presented by Manku et al. [20] performs in O(_klog¹ _k(logn)²) ex- pected number of steps, where k is the number of long range links per node in the mesh. Assuming some extra knowledge on the long range links, Fraigniaud et al. [12] described an oblivious routing which performs inO((logn)^1+1/d) expected number of steps. Finally, Martel and Nguyen [21] presented a non oblivious routing protocol achieving the same performances under the same assumption as in [12].

1.3 Organization of the paper

The paper is organized as follows : Section 2 presents the main result of the paper by exhibiting the non navigability of graphs with doubling dimensionlog logn. In Section 3, we study the special case of square meshes, and prove that they are all navigable.

2 Non navigable graphs

In this section, we prove that the result in [24] is essentially the best that can be achieved as far as doubling dimension is concerned.

Theorem 1 Letδ:N7→N be such that limn→∞log logn

δ(n) = 0. ThenG^δ is not navigable.

Informally, the argument of the proof is that a doubling dimensionlog lognimplies that the number of possible “directions” where a random link can go is greater than any polylogarithm ofn. Therefore, for any trial of the long range links, there always exist a direction for which these long links do not help in the sense that there exist a source and a target between which greedy routing does not use any long range link.

Proof. We show that there exists an infinite family of graphs {G⁽ⁿ⁾, n≥1}indexed by their number of vertices, such that G⁽ⁿ⁾ ∈ Gn,δ(n) and for any family Φ = {ϕ⁽ⁿ⁾, n≥1}of collections of probability distributions, greedy routing in (G⁽ⁿ⁾, ϕ⁽ⁿ⁾) performs in an expected number of stepst(n)∈/O(polylog(n)) for some pairs of source and target.

Let d : N 7→ N be such that d ≤ δ, limn→∞log logn

d(n) = 0, and d(n) ≤ ε√

logn for some 0 <

ε < 1. For the sake of simplicity, assume that p = n^1/d(n) is integer. G⁽ⁿ⁾ is the graph of n nodes consisting of p^d(n) nodes labeled (x1, . . . , xd(n)), xi ∈Zp. Node (x1, . . . , xd(n)) is connected to all nodes (x1+a1, . . . , xd(n)+ad(n)) whereai ∈ {−1,0,1},i= 1, . . . , d(n), and all operations are taken modulo p(cf. Figure 1). Note that, by construction ofG⁽ⁿ⁾, the distance between two nodesy = (y1, . . . , yd(n)) andz= (z1, . . . , zd(n)) is max1≤i≤d(n)min(|yi−zi|, p− |yi−zi|). Hence, the diameter of G⁽ⁿ⁾isbp/2c. Claim 1 G⁽ⁿ⁾∈ G^n,δ(n).

dir(+1,+1)

dir(+1,0) dir(−1,0)

dir(0,+1)

dir(0,−1) dir(−1,−1)

dir(−1,+1)

dir(+1,−1)

Fig. 1 – Example of graphG⁽ⁿ⁾ defined in proof of Theorem 1 withd(n) = 2. Grey areas represent the variousdirections for the central node. The bold line represents adiagonal for the central node. Colored nodes belongs to aline.

ClearlyG⁽ⁿ⁾ hasnnodes. We prove thatG⁽ⁿ⁾ has doubling dimensiond(n), therefore at most δ(n).

Let 0 = (0, . . . ,0). The ball B(0,2r) can be covered by 2^d(n) balls of radius r, centered at the 2^d(n) nodes (x1, . . . , xd(n)), xi ∈ {−r,+r}for any i = 1, . . . , d(n). Hence the doubling dimension ofG⁽ⁿ⁾ is at mostd(n). On the other hand, |B(0,2r)| = (4r+ 1)^d(n) and |B(0, r)| = (2r+ 1)^d(n). Thus at least (4r+ 1)^d(n)/(2r+ 1)^d(n) balls are required to coverB(0,2r), since in G⁽ⁿ⁾, for any node uand radius r, |B(u, r)| = |B(0, r)|. This ratio can be rewritten as 2^d(n)(1− 2(2r+1)¹ )^d(n). For 2r = ^n1/d(n)₅ , since d(n)≤√logn, we get that (2r+ 1)> ²

√logn

5 > d(n) forn≥n0,n0≥1. Then, forn≥n0,

1− 1

2(2r+ 1) d(n)

>

1− 1

2d(n) d(n)

= 2^{d(n) log}(¹−2d(n)¹ )

≥2^d(n)

“

−2d(n)¹ −_4(d(n))2⁴ ”

= 2^{−12−d(n)1}

There exists n1 ≥ n0, such that 2^{−12−d(n)1} > ¹₂ for n ≥ n1. Then, for n ≥ n1, |B(0,2r)|/|B(0, r)| >

2^d(n)−1. Thus the doubling dimension ofG⁽ⁿ⁾ is at leastd(n), which proves the claim.

Definition 1 For any node u= (u1, . . . , ud(n)), and for any D = (ν1, . . . , νd(n))∈ {−1,0,+1}^d(n), we call direction the set of nodes

diru(D) ={v= (v1, . . . , vd(n)) : vi= (ui+νi·xi) modp,1≤xi≤ bp/2c}.

Note that, for anyu, the directions diru(D) forD∈ {−1,0,+1}^d(n)partition the nodes of G⁽ⁿ⁾(see Figure 1). There are obviously 3^d(n)directions, and the 2^d(n)directions defined on{−1,+1}^d(n)have all the same cardinality.

Definition 2 For any nodeu= (u1, . . . , u_d(n)), and for any D= (ν1, . . . , ν_d(n))∈ {−1,+1}^d(n), we call diagonalthe set of nodes

diag_u(D) ={v= (v1, . . . , vd(n)) : vi= (ui+νi·x) modp,1≤x≤ bp/2c}.

The next claim shows that long range links are useless for greedy routing along a diagonal if they are not going in the direction of the diagonal.

3

Claim 2 Let u be any node and let v be the long range contact of ufor some distribution ϕ⁽ⁿ⁾ of the long range links. Assumev ∈diru(D)and let t∈diag_u(D⁰) for D, D⁰ ∈ {−1,+1}^d(n), D 6=D⁰. Greedy routing from utotdoes not use the long range link(u, v).

Let u= (u1, . . . , ud(n)), v = (v1, . . . , vd(n)) andt = (t1, . . . , td(n)). Since t ∈diag_u(D⁰), there exists x∈ {1, . . . ,bp/2c}such that|ti−ui|=xfor all 1≤i≤d(n). SinceD6=D⁰, there existsj∈ {1, . . . , d(n)} such that :

tj=uj+α·x (1)

vj =uj−α·y, (2)

for someα∈ {−1,+1}andy∈ {1, . . . ,bp/2c}. Then,

dist(v, t)≥ |tj−vj|=x+y > x= dist(u, t).

Therefore greedy routing fromutotdoes not use the long range link (u, v), which proves the claim.

Consider now a distributionϕ⁽ⁿ⁾of long range links that belongs to some given collection of probability distributions Φ ={ϕ⁽ⁿ⁾, n≥1}. We prove that routing on the diagonal is hard. More precisely, letsbe a source node and t be a target node, t ∈ diag_s(D) for some D. If any node u∈ diag_s(D) between s and thas its long range contactv ∈diru(Dv) for someDv 6=D, then, from the previous claim, greedy routing froms tot does not use any long range link and thus takes dist(s, t) steps. We prove that this phenomenon occurs for at least one pair (s, t) such that dist(s, t)≥2^d(n)−3.

Definition 3 An intervalI = [a, b]is a connected subset of a diagonal. Precisely,[a, b] is an interval of diag_a(D)ifb∈diag_a(D) and

I ={c∈diag_a(D)|dist(a, c) + dist(c, b) = dist(a, b)}.

We say that an intervalI of diag_u(D) isgood if there existsx∈I such that the long range contact y ofxsatisfiesy∈dirx(D).

Definition 4 A lineL ofG⁽ⁿ⁾ in direction D∈ {−1,+1}^d(n) is a maximal subset ofV(G⁽ⁿ⁾)such that for any two nodes u, v∈L, we have

diag_u(D)∩diag_v(D)6=∅.

The set of all the lines in the same directionD partitionsG⁽ⁿ⁾inton/plines of size p.

Let us partition each line intop/Xdisjoint intervals of same lengthX. This results inton/Xintervals per direction, thus in total into a setSof _Xⁿ ·2^d(n)intervals of lengthX, since there are 2^d(n)directions defined in{−1,+1}^d(n). We show that ifX is too small, then there is at least one of all the intervals in S which is not good.

There is a one-to-one mapping between intervals and nodes in the following sense. Each good interval I = [a, b]∈ S must contain a node uwhose long range contact v satisfiesv ∈diru(D). The node u is called thecertificate forI. Nodeucannot be the certificate of any other intervalJ ∈S withJ 6=I, even for those such thatJ∩I 6=∅whenI andJ are in two distinct directions.

We have 2^d(n)· Xⁿ intervals in S. Since a certificate certifies the goodness of exactly one interval, 2^d(n)·Xⁿ has to be at mostn, that is :X ≥2^d(n). By the pigeonhole principle, ifX <2^d(n), there is one intervalI = [s, t]∈S which is not good. From Claim 2, greedy routing fromsto ttakesX−1 steps.

Sinced(n)≤ε√

logn, we have :

p=n^1/d(n)≥2^1ε√logn≥2^ε√logn−2≥2^d(n)−2.

Therefore, the value X = 2^d(n)−2 can be considered for our partitioning. In this case, we obtain that greedy routing fromsto ttakes 2^d(n)−3 steps.

We complete the proof of the theorem by proving the following claim.

Claim 3 2^d(n)∈/O(polylogn).

Letα≥1, we have :

log^αn

2^d(n) = 2^{αlog logn−d(n)}= 2^{α d(n)}(^{log log}d(n)ⁿ−_α¹). Since limn→∞log logn

d(n) = 0, there existsn1≥1 such that for anyn≥n1,_{log logn}

d(n) −α¹

≤ −2α¹ , and thus

log^αn

2^d(n) ≤2^−d(n)/2. Moreover,d(n)≥log logn, then, forn≥n1, log^αn

2^d(n) ≤2^{−log log2 n} =o(1).

In other words, 2^d(n)is not a polylogarithm ofn, which proves the claim.

Remark. Note that the proof of Theorem 1 does not assume independent trials for the long range links.

3 Navigability of meshes

The family of non navigable graphs defined in the proof of Theorem 1 contains the standard square meshes of dimension d(n) as subgraphs, where d(n) log logn. Nevertheless, and somehow counter intuitively, a supergraph of a navigable graph is not necessarily navigable. In this section, we illustrate this phenomenon by focusing on the special case of d-dimensional meshes, the first graphs that were considered for the analysis of navigable networks [17]. Precisely, we show that anyd-dimensional torus C_n^1/d×. . . C_n^1/dis navigable : either it has a polylogarithmic diameter, or it admits a distribution of links such that greedy routing computes paths of polylogarithmic length. This result has partially been proven in [9] for the case of constant dimensions. We give here a complete proof that holds for any dimension.

Theorem 2 For any positive functiond(n), the n-noded(n)-dimensional torus is navigable.

Proof. We construct a random link distribution ϕ as follows. Let u = (u1, . . . , ud(n)) and v = (v1, . . . , vd(n)) be two nodes. If they differ in more than one coordinate, then ϕu(v) = 0 ; otherwise, i.e. they differ in only one coordinate, say theith, then :

ϕu(v) = 1 d(n)· 1

2Hk · 1

|ui−vi|, wherek=^n1/d₂ andHk=Pk

j=11

j is the harmonic sum. Note that this distribution corresponds to : – picking a dimension uniformly at random (probability _d(n)¹ to pick dimensioni)

– and to draw a long-range link on this axis according to the 1-harmonic distribution over distances (_2H¹_k is the normalizing coefficient for this distribution), which is the distribution given by Kleinberg to make the 1-dimensional torus navigable.

Let nows= (s1, . . . , sd(n)) andt= (t1, . . . , td(n)) be a pair of source and target in the mesh. Assume that the current message holder during an execution of greedy routing isx= (x1, . . . , xd(n)), at distanceX fromt. The probability thatxhas a long range link to some nodew= (x1, . . . , xi−1, wi, xi+1, . . . , xd(n)), 1 ≤ i ≤ d(n) such that |ti −wi| ≤ |ti −xi|/2, is at least P

1≤i≤d(n) 1

3d(n)Hk = _3H¹

k, along the same analysis as the analysis of Kleinberg one dimensional model, summing over the dimensions. If such a link is found, it is always preferred to the local contact of x that only reduces one of the coordinate by 1.

Thus, after at most 3Hksteps on expectation, one of the coordinates has been divided by two. Note that since long range links only get to nodes that differs in a single coordinate from their origin, further steps cannot increase|xi−ti|for any 1≤i≤d(n),xbeing the current message holder. Repeating the analysis for all coordinates, we thus get that after 3d(n)Hk steps on expectation, all the coordinates have been divided by at least two, and so the current distance to the target is at mostX/2. Finally, the algorithm reachestafter at most 3d(n)Hklog(dist(s, t)) steps on expectation, which isO (log²n)/d(n)

.

Remark. Note that our example of non-navigability in Section 3 may appear somehow counter intuitive in contrast to our latter construction of long range links on meshes. Indeed, why not simply repeating such a construction on the graph G⁽ⁿ⁾ defined in the proof Theorem 1 ? That is, why not

5

Fig.2 – Example of 2-dimensional mesh augmented as in proof of Theorem 2 : 1-harmonic distribution of links on each axis. Bold links are long range links, they are not all represented.

selecting long range contacts on each ”diagonal” using the 1-harmonic distribution, in which case greedy routing would perform efficiently between pairs (s, t) on the diagonals ? This cannot be done however because, to cover all possible pairs (s, t) on the diagonals, 2^d(n) long range links per node would be required, which is larger than any polylogarithm ofnwhend(n)log logn.

4 Conclusion

The increasing interest in graphs and metrics of bounded doubling dimension arises partially from the hypothesis that large real graphs do present a low doubling dimension (see, e.g., [10, 16] for the Internet). Under such an hypothesis, efficient compact routing schemes and efficient distance labeling schemes designed for bounded doubling dimension graphs would have promising applications. On the other hand, the navigability of a network is actually closely related to the existence of efficient compact routing and distance labeling schemes on the network. Indeed, long range links can be turned into small labels, e.g. via the technique of rings of neighbors [24]. Interestingly enough, our paper emphasizes that the small doubling dimension hypothesis of real networks is crucial. Indeed, for doubling dimension above log logn, networks may become not navigable. It would therefore be important to study precisely to which extent real networks do present a low doubling dimension.

In a more general framework, our result of non navigability shows that the small world phenomenon, in its algorithmic definition of navigability, is not only due to the good spread of additional links over distances in a network, but is also highly dependent of the base metric itself, in particular in terms of dimensionality.

Peleg recently proposed the more general question of f-navigability. For a functionf, we say that a n-node graph Gisf-navigable if there exists a distribution ϕof long range links such that the greedy diameter of the augmented graph (G, ϕ) is at mostf(n). From [23], alln-node graphs are√n-navigable by giving an uniform random distribution of the long range links. From Theorem 1, we get as a corollary that, for all graphs to bef-navigable,f(n) = Ω(2^√logn). It thus remains to close the gap between these upper and lower bounds for thef-navigability of arbitrary graphs.

Acknowledgments. We are thankful to Ph. Duchon for having pointed to us an error in an earlier version of this paper.

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